Ruled surfaces and developable surfaces Johannes Wallner, Graz University of Technology Ruled surfaces and developable surfaces: The “waves” sculpture by Santiago Calatrava — The bunny approximated by a piecewise-developable surface — a curved-folding design Erik and Martin Demaine. Contents 1 Ruled surfaces and developable surfaces2 1A Representations of ruled surfaces..........2 1B Intrinsically flat surfaces...............3 1C Relation between developability and ruledness...4 1D Developables with creases..............5 2 Modeling with ruled surfaces7 2A B-spline curves and surfaces.............7 2B Modeling ruled surfaces using optimization.....8 2C Modeling capabilities of ruled surfaces....... 10 3 Developables in classical surface theory 13 3A Conjugate nets.................... 13 3B Developables in support structures......... 14 3C Design dilemmas.................. 15 4 Modeling with developable surfaces 17 4A Convex developables................ 17 4B Developables via their dual representation..... 18 4C Developables as quadrilateral meshes........ 19 4D Spline techniques.................. 20 4E Developables as triangle meshes........... 21 4F Modeling curved folds................ 21 References 24 1 Ruled surfaces and developable surfaces 1A Representations of ruled surfaces Primal representation. Ruled surfaces are traced out by the movement of a straight line through space, and they are usually de- scribed by a correspondence between parametric curves a(u) and b(u): The parametric description of a ruled surface is x(u; v) = (1 − v)a(u) + vb(u) = a(u) + vr(u); where r(u) = b(u) − a(u): For fixed u, the expression x(u; v) describes the straight line which connects points a(u) and b(u). It is called a ruling. This represen- tation of ruled surfaces is referred to as the primal one, in order to distinguish it from the dual representation which is defined later. FIGURE 1.2: Sculpture by Santiago Calatrava. One can clearly see that when a point progresses along a ruling of a ruled surface, the tangent plane in that point rotates abou the ruling. The total rotation is 180 degrees. Definition 1.2 A ruled surface where all rulings have only 1 single tangent plane is called a torse, or a developable ruled surface. Example 1.3 General cylinders and cones are torses, and so are the surfaces traced out by the tangents of a space curve a(u). The easy proofs of these statements have been relegated to Exer- cises 1.1 and 1.5 (page6). FIGURE 1.1: Ruled surface defined by the correspondence between two curves a(u), b(u). Much of the behaviour of ruled surfaces is governed by the rota- tion of the tangent plane as one moves along a ruling. Either that motion is a complete rotation, or there is no rotation at all. This fact is responsible for the modeling capabilities, and the modeling restrictions a designer is faced with. Lemma 1.1 Consider the ruling R(u) = a(u) _ b(u) of the sur- FIGURE 1.3: Tangent surfaces: The surface traced out by the tan- face defined by the correspondence between curves a(u), b(u). gents of a curve c(u) is a developable ruled surface. The curve 1. If the tangent plane of the surface is different in 2 different itself is a sharp edge on the surface. Here only one half of each points of R(u), it rotates about 180 degrees when R(u) is tangent is shown. traversed along its entire length (there are no singular points on R(u)). Dual representation. Another way of describing a time-depen- 2. If the tangent plane is the same in 2 different points of R(u), dent straight line R(u) is via the envelope of a moving plane T (u): it is the same for all points of R(u) (there may be 1 singular > “regression” point without tangent plane on R(u)). T (u) ::: n x + n0 = 0 where n = (n1; n2; n3) 3. If two points are without tangent plane, the entire ruling is. and each of n0; : : : ; n3 is a function of u. Since the intersection of These three cases are characterized by the vectors two successive planes T (u) and T (u + h) is a straight line, this is also true for the limit h ! 0, and it is not difficult to compute the a_ ; b_ ; b − a; or equivalently, a_ ; r_; r rulings of the surface enveloped by the family T (u): spanning a subspace of dimensions 3, 2, and 1 respectively. R(u) = lim T (u) \ T (u + h): h!0 Proof: A normal vector of the surface is computed by That ruling is not difficult to compute, since the condition that x 2 xu × xv = (a_ + vr_) × r = a_ × r + vb_ × r: T (u) \ T (u + h) can be modified as follows: _ > > Either the two vectors a_ × r, b × r are both zero (case 3) or are n(u) x + n0(u); n(u + h) x + n0(u + h) () parallel (case 2) or are not parallel (case 1). Correspondingly the 8 > normal vector is zero (case 3), or does not change its direction but <n(u) x + n0(u); may vanish for 1 value of v (case 2) or rotates by 180 degrees with- () n(u + h) − n(u) > n0(u + h) − n0(u) : x + = 0: out ever vanishing (case 1). Q.E.D h h The limit h ! 0 now yields the conditions > > R(u) ::: n x + n0 = n_ x +n _ 0 = 0: T (u + h) r b The ruling R(u) is the common solution of these two equations. T (u) a The vector indicating the direction of the ruling is accordingly com- c h) a_ puted as (u − r = n × n_ : T The surface traced out by the lines R(u) is ruled, and obviously the tangent plane of the surface along the entire ruling is the plane T (u). The representation of ruled surfaces as an evelope of planes works only for torses (developable ruled surfaces), and is called the FIGURE 1.5: Developable ruled surface defined as the enve- dual represention. lope of a family T (u) of planes. The rulings occur as limits of T (u) \ T (u + h) as h ! 0 (i.e., a ruling is the intersection of Discrete ruled surfaces. Both for computations and for discrete infinitesimally close planes). The points of regression c(u) occur theories (e.g. discrete differential geometry) it makes sense to study as limits of T (u − h) \ T (u) \ T (u + h) as h ! 0 (i.e, a regres- discrete representations of ruled surfaces. While a general ruled sion point is the intersection of 3 infinitesimally close planes, or 2 surface is simply a sequence of lines or a sequence of straight line infinitesimally close rulings). segments, the condition of developability is best expressed by re- quiring that successive lines or successive line segments are co-pla- nar (see Figure 1.4). Figure 1.5 shows a discrete model of a developable ruled surface. It suggests properties which are known to be true for continuous developable surfaces, namely the existence of a curve c(u) of sin- gular points, and the fact that the tangents of that curve are exactly the rulings of the developable surface. 1B Intrinsically flat surfaces Developable surfaces constitute a class of surfaces whith many in- FIGURE 1.6: Surfaces created by isometric bending of a rectangu- teresting properties relevant to different kinds of applications. Un- lar sheet of paper (images by Solomon et al. [2012]). The left hand fortunately the mathematical statements which express the relations surface consists of a planar part and 4 individual ruled parts. between these defining properties are complicated. Developable surfaces are notorious for statements which are not true in the math- ematical sense but are nevertheless true for all practical purposes. By gluing 2 opposite edges of a rectangle together we obtain a met- ric space which is isometric to a right circular cylinder; by cutting We have already defined developability as a special property of a right circular cylinder along a ruling yields a surface which can ruled surfaces. This word comes from the fact that such devel- be isometrically mapped to a rectangle. Therfore the right circular opables can be unfolded into the plane without stretching or tearing, cylinder is an intrisically flat surface. and in a manner of speaking, also the converse statement is true. This unfoldability is the more literal meaning of “developable”. It is One can also glue together the remaining 2 opposite edges of a however convenient to require this property only in a weaker sense, cylinder and ask the question if there exists a surface in 3-space because we want to be able to call cylinders developable, and a which is isometric to this intrinsically flat Riemannian manifold. cylinder can only be unfolded if it is first cut open. This question was answered affirmative by John Nash via his fa- mous embedding theorem: Definition 1.4 A surface is intrinsically flat (“developable” in the literal sense), if every point has a neighbourhod which can be mapped to a planar domain in an isometric way, meaning that curves within the surface do no change their length. FIGURE 1.4: A discrete torse is formed by a sequence of line FIGURE 1.7: The cylindrical part of a tin can is a right circular segments such that each seg- cylinder, and therefore intrinsically flat. This property is not lost ment and its successor are co- upon isometric deformation. planar. Theorem 1.5 (J. Nash 1954) If M is an m-dimensional Rieman- nian manifold, then there is a C1 surface in n isometric to M, R FIGURE 1.9: A flat provided n > m and there is a surface in n diffeomorphic to M R polyhedral torus.